Workers at Companies X and Y are paid the same base hourly rate. Workers at company X are paid 1.5 times the base hourly rate for each hour worked per week in excess of the first 37, while workers at Company Y are paid 1.5 times the base hourly rate for each hour worked per week in excess of the first 40. In a given week, how many hours must a Company X worker work in order to receive the same pay as a company Y worker who works 46 hours?
Show Hint
- 46
- 45
- 44
- 43
- 42
The Correct Option is B
Solution and Explanation
This is a word problem that requires setting up algebraic expressions for the total weekly pay of workers at two different companies and then equating them to solve for an unknown number of hours.
Step 2: Detailed Explanation:
Let \(R\) be the base hourly rate, which is the same for both companies.
Calculate the pay for the Company Y worker:
The worker works 46 hours. Overtime at Company Y starts after 40 hours.
- Regular hours: 40 hours
- Overtime hours: \(46 - 40 = 6\) hours
- Regular pay: \(40 \times R = 40R\)
- Overtime rate: \(1.5 \times R = 1.5R\)
- Overtime pay: \(6 \times 1.5R = 9R\)
- Total pay for Company Y worker: \(40R + 9R = 49R\)
Set up the pay for the Company X worker:
Let \(H\) be the number of hours the Company X worker must work. Overtime at Company X starts after 37 hours.
We want the Company X worker's pay to equal \(49R\).
- Regular pay for the first 37 hours: \(37 \times R = 37R\)
- Overtime hours: \(H - 37\) hours
- Overtime pay: \((H - 37) \times 1.5R\)
- Total pay for Company X worker: \(37R + (H - 37) \times 1.5R\)
Equate the two pays and solve for H:
\[ \text{Pay}_X = \text{Pay}_Y \] \[ 37R + (H - 37) \times 1.5R = 49R \] Since \(R\) is a non-zero base rate, we can divide the entire equation by \(R\):
\[ 37 + (H - 37) \times 1.5 = 49 \] Now, solve for \(H\):
\[ (H - 37) \times 1.5 = 49 - 37 \] \[ (H - 37) \times 1.5 = 12 \] \[ H - 37 = \frac{12}{1.5} \] Since \(\frac{12}{1.5} = \frac{12}{3/2} = 12 \times \frac{2}{3} = 8\):
\[ H - 37 = 8 \] \[ H = 8 + 37 = 45 \] Step 3: Final Answer:
The Company X worker must work 45 hours.
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